Heat transfer and thermal stress during sapphire crystal growth using the heat exchanger method is numerically studied based on two radiation models: the no-slip Rosseland diffusion approximation and the finite volume method (FVM). The results indicate that at the full-diameter growth stage the Rosseland model gives almost the same melt-crystal interface shape and stream function for melt flow as the rigorous FVM. However, it overpredicts the heat transfer through the sapphire domain and fails completely to capture the steep variation in temperature, and the consequent high temperature gradient and thermal stress in the narrow bottom region of the crystal. The temperature gradient and thermal stress in this region predicted by the Rosseland approximation gradually approach the FVM predictions as the optical thickness is increased. However, disparities remain even at a very large optical thickness.